When representing the frequency of different results in our data, we often choose to use a frequency table.
A frequency table communicates the frequency of each result from a set of data. This is often represented as a column table with the far-left column describing the result and any columns to the right recording frequencies of different result types.
Frequency tables can help us find the least or most common results among categorical data. They can also allow us to calculate what fraction of the data a certain result represents.
When working with numerical data, frequency tables can also help us to answer other questions that we might have about how the data are distributed.
For example, the following list of colours were recorded in the frequency table below:
\text{White, Black, White, Black, Black, Blue, Blue, White, Red, White,}\\ \text{ White, Blue, Orange, Blue, White, White, Orange, Red, Blue, Red}
Car colour | Frequency |
---|---|
\text{Red} | 3 |
\text{Black} | 3 |
\text{White} | 7 |
\text{Orange} | 2 |
\text{Blue} | 5 |
We can find the mode, mean, median and range from a frequency table. These will be the same as the mode, mean, median and range from a list of data but we can use the frequency table to make it quicker.
The cumulative frequency is the sum of the frequencies of the score and each of the scores below it. The cumulative frequency of the first row will be the frequency of that row. For each subsequent row, add the frequency to the cumulative frequency of the row before it.
Consider the following data set.
Score | Frequency | Cumulative frequency |
---|---|---|
2 | 3 | 3 |
3 | 5 | 8 |
4 | 3 | 11 |
5 | 4 | 15 |
6 | 8 | 23 |
7 | 2 | 25 |
How many scores are there in total?
Find the median score.
We want to find the mean of the following data set.
\text{Score }(x) | \text{Frequency }(f) | xf |
---|---|---|
2 | 7 | 14 |
3 | 2 | 6 |
4 | 8 | 32 |
5 | 5 | 25 |
6 | 4 | 24 |
7 | 7 | 49 |
How many scores are there in the data set?
What is the total sum of all the scores in the data set?
Find the mean for this data set.
We can use the frequency table to find the mean, mode, median, and range of a data set.
The cumulative frequency is the sum of the frequencies of the score and each of the scores below it. Adding a cumulative frequency column to a frequency table is helpful for finding the median.
Adding an xf column to a frequency table is helpful for finding the mean.
When the data are more spread out, sometimes it doesn't make sense to record the frequency for each separate result and instead we group results together to get a grouped frequency table.
A grouped frequency table combines multiple results into a single group. We can find the frequency of a group by adding all the frequencies of the results contained in that group.
The modal class in a grouped frequency table is the group that has the greatest frequency. If there are multiple groups that share the greatest frequency then there will be more than one modal class.
The drawback of a grouped frequency table is that the data becomes less precise, since we have grouped multiple data points together rather than looking at them individually.
Complete the frequency table for the data set below.
77,\,54,\,53,\,56,\,73,\,55,\,94,\,95,\,76,\,52,\,72,46,\,85, \\61,\,48,\,90,\,64,\,70,\,40,\,52,\,57,\,88,\,59,\,95,\,61
Class | Frequency | Cumulative frequency |
---|---|---|
40-49 | ||
50-59 | ||
60-69 | ||
70-79 | ||
80-89 | ||
90-99 |
A grouped frequency table combines multiple results into a single group. We can find the frequency of a group by adding all the frequencies of the results contained in that group.
The modal class in a grouped frequency table is the group that has the greatest frequency. If there are multiple groups that share the greatest frequency then there will be more than one modal class.
When finding the mean and median of grouped data we want to first find the class centre of each group. The class centre is the mean of the highest and lowest possible scores in the group.
We want to estimate the median for this data set.
\text{Class} | \text{Frequency} | \text{Cumulative frequency} |
---|---|---|
21 - 25 | 4 | 4 |
26 - 30 | 3 | 7 |
31 - 35 | 3 | 10 |
36 - 40 | 2 | 12 |
41 - 45 | 5 | 17 |
46 - 50 | 8 | 25 |
How many scores are there in total?
Estimate the median.
Estimate the mean for this data set. Round your answer to one decimal place.
\text{Class} | \text{Class centre } (x) | \text{Frequency } (f) | xf |
---|---|---|---|
6 - 10 | 8 | 2 | 16 |
11 - 15 | 13 | 1 | 13 |
16 - 20 | 18 | 9 | 162 |
21 - 25 | 23 | 8 | 184 |
26 - 30 | 28 | 7 | 196 |
31 - 35 | 33 | 5 | 165 |
When estimating the mean and median of grouped data we use the class centre of each group.
The class centre is the mean of the highest and lowest possible scores in the group.